Semilocal Pauli–Gaussian Kinetic Functionals for Orbital-Free Density

Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 4385−4390

pubs.acs.org/JPCL

Semilocal Pauli−Gaussian Kinetic Functionals for Orbital-Free Density Functional Theory Calculations of Solids Lucian A. Constantin,† Eduardo Fabiano,†,‡ and Fabio Della Sala*,†,‡ †

Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy Institute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, Italy

‡

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S Supporting Information *

ABSTRACT: Kinetic energy (KE) approximations are key elements in orbital-free density functional theory. To date, the use of nonlocal functionals, possibly employing system-dependent parameters, has been considered mandatory in order to obtain satisfactory accuracy for different solid-state systems, whereas semilocal approximations are generally regarded as unfit to this aim. Here, we show that, instead, properly constructed semilocal approximations, the Pauli−Gaussian (PG) KE functionals, especially at the Laplacian level of theory, can indeed achieve similar accuracy as nonlocal functionals and can be accurate for both metals and semiconductors, without the need for system-dependent parameters.

T

he kinetic energy (KE) functional is a fundamental quantity in electronic structure theory. It plays a prominent role in subsystem and embedding theories,1−5 hydrodynamic models,6−8 information theory,9,10 machine learning techniques for Fermionic systems,11−13 potential functional theory,14 various extensions of the Thomas−Fermi (TF) theory,15,16 and especially orbital-free density functional theory (OF-DFT).17−23 The applicability of these methods is strongly limited by the lack of accurate and simple KE approximations. OF-DFT allows an efficient description of the ground state of any electronic system via the solution of the Euler equation24,25 δTs[n] + vext(r) + δn

′

∫ |rn−(r )r′| d3r′ +

δExc[n] =μ δn

On the other hand, unlike for the exchange−correlation energy case where semilocal approximations26,27,38 have experienced huge success in the context of Kohn−Sham (KS) DFT,39 semilocal KE approximations, which scale linearly with system size and can be easily implemented in both real-space and wave-vector formalisms, are barely used in OF-DFT calculations. This traces back to the fact that the actual state-of-the-art semilocal functionals (e.g., VT84f,40 vWGTF1/vWGTF241) may encounter severe failures for various systems (e.g., semiconductors) and properties (e.g., bulk modulus, vacancy energy41). In this Letter, we show that this limitation is not a fundamental feature of the semilocal functionals but is just related to the approximations employed so far. Indeed, we show that even a simple combination of the local TF42,43 and gradient-dependent von Weizsäcker (W)44 functionals can outperform the actual state-of-the-art semilocal functionals in solid-state calculations. Moreover, we show that a simple nonempirical Laplacian-level semilocal KE functional can easily approach the accuracy of nonlocal KE approximations, achieving broad accuracy and applicability. These results shed completely new light on the topic of KE functionals, showing that with careful development semilocal KE functionals can be applied with good accuracy in large-scale OFDFT applications. To this purpose, we have performed OF-DFT calculations for simple metals41 (Li, Mg, and Al in simple-cubic (sc), facecentered-cubic (fcc), and body-centered-cubic (bcc) configurations) and III−V semiconductors36,37 (AlP, AlAs, AlSb,

(1)

where n is the ground-state electron density, Ts is the noninteracting KE functional, vext is the external (e.g., nuclear) potential, Exc is the exchange−correlation functional,26,27 and μ is the chemical potential. The vast majority of the OF-DFT calculations employ nonlocal (or two-point) KE functionals that are rather accurate and display a logarithmic 6(N ln(N )) scaling behavior with system size N. However, these functionals are based on the Lindhard response function of the noninteracting homogeneous electron gas (HEG).28−33 Thus, most of them depend on the average density in the unit cell n0, and therefore, they are not adequate for finite systems or even for anisotropic solid-state systems such as interfaces, surfaces, or layered materials, where n0 may not be well-defined34−36 or may not be representative of the system. Moreover, very accurate results are often obtained only using system-dependent parameters.32,33,36,37 © XXXX American Chemical Society

Received: June 20, 2018 Accepted: July 18, 2018 Published: July 18, 2018 4385

DOI: 10.1021/acs.jpclett.8b01926 J. Phys. Chem. Lett. 2018, 9, 4385−4390

Letter

The Journal of Physical Chemistry Letters

HC for semiconductors. In contrast, the semilocal functionals (TF(1/5)W, TFW, VT84f) perform much worse (VT84f is quite accurate only for metals). Next we consider more general semilocal functionals. Any Generalized Gradient Approximation (GGA) or Laplacianlevel meta-GGA semilocal KE functional that behaves correctly under uniform density scaling (i.e., Ts[nγ(r)] = γ2Ts[n(r)],46,47 where nγ(r) = γ3n(γr), with γ ≥ 0) can be written

GaP, GaAs, GaSb, InP, InAs, and InSb with a cubic zinc-blende unit cell), comparing our results to KS-DFT values obtained using the same computational setup. These systems have been largely employed to assess the accuracy of nonlocal KE functionals.32,35−37,41 We considered four properties: cell volume (V0), bulk modulus (B), total energy at equilibrium volume (E0), and density error (D0). The first three properties have been previously considered in the assessment of functionals.32,35−37,41 The last quantity is defined as 1 D0 = |nKS(r) − nOF‐DFT(r)| d3r Ne (2)

Ts[n] = TsW +

∫

∑ p

2

5 Fs,pλ = 1 + (λ − 1) s 2 3

MAREp HC (1/2)(MARESM p + MARE p )

(4)

where τ = (3/10)kF n is the TF KE density, with kF = (3π2n)1/3 being the Fermi wave vector, s = |∇n|/(2kFn) and q = ∇2n/(4kF2n) are the reduced gradient and Laplacian, TW s = ∫ τTF(5s2/3) d3r is the W44 KE (which is exact for one- and two-electron systems, as well as for any bosonic system), and Fp is the Pauli KE enhancement factor.24 The exact condition p Ts ≥ TW s requires that F > 0. However, this constraint is satisfied only by a few semilocal approximations,40,41,48 and it is even violated by some nonlocal KE functionals.49 Rehabilitation of Semilocal KE Functionals for OF-DFT. In a first attempt, we considered the family of functionals TFλW, which are defined by TF

and is computed with the KS lattice constant for both KS and OF-DFT. Here, Ne is the electron number in the unit cell. The density error is a very hard test for the quality of the KE functional, describing how well the OF-DFT calculations converge to the exact density. For each quantity p ∈ {V0, B, E0, D0}, we considered the mean absolute relative error (MAREp) with respect to the reference KS values averaging over all systems (metals or semiconductors). Finally, in order to have a global indicator for all of the properties, we considered a relative MARE (RMARE) obtained normalizing to the average values of the Smargiassi and Madden (SM)30 and the Huang and Carter (HC)37 functionals, i.e. RMARE =

∫ τ TFFsp(s , q) d3r 42,43

(5)

This class of functionals has been investigated for atomic/ molecular systems50,51 but not for bulk systems with pseudopotentials. The performance for various values of λ is reported in Figure 1. The first interesting result of this Letter is that the RMAREs can be strongly reduced by varying λ, reaching for λ = 0.6 RMARE ≈ 1.2 and 1.9 for metals and semiconductors, respectively. Thus, a very simple functional, TF(0.6)W, is already better than the current semilocal state-ofthe-art (VT84f). However, the Pauli enhancement factor of TFλW becomes negative whenever s2 ≥ 0.6/(1−λ). The relevant values of s2 for solids are comprised in the interval [0:1], as shown in Figure 2 by s-decomposition of the TF KE energy t[n](s), so that52

(3)

The HC (with universal parameters λ = 0.01177 and β = 0.714336) and SM functionals have been chosen as references because they bind both metals and semiconductors33 and do not employ system-dependent parameters (actually, HC does not even depend on the average density n0). Thus, a functional with RMARE = 1 shows performance between HC and SM. Figure 1 reports the RMAREs for metals and semiconductors of different semilocal functionals commonly used

Ts[n] =

∫0

+∞

ds t[n](s)Fs(s)

(6)

Figure 1. RMARE of metals vs RMARE of semiconductors for different functionals.

in OF-DFT calculations, namely, VT84f,40 TFW, TF(1/5)W (the latter being combinations of the TF functional with W and W/5, respectively), as well as for the nonlocal references HC and SM. Note that other functionals from literature, not only semilocal, but also WT,28 WGC,45 and vWGTF,41 do not bind semiconductors;33,36 therefore, they cannot be included in Figure 1. Clearly, the nonlocal functionals give the lowest RMAREs, with SM being especially accurate for metals and

Figure 2. Upper panel: Pauli enhancement factor for different functionals. Lower panel: s-decomposition of the TF KE for different systems. 4386

DOI: 10.1021/acs.jpclett.8b01926 J. Phys. Chem. Lett. 2018, 9, 4385−4390

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The Journal of Physical Chemistry Letters

χ(η) = (kF/π2)(1/FLind(η)), with η = k/2kF and FLind being the dimensionless wave vector and the Lindhard function,57 respectively. Figure 3 reports the behavior of the linear

Thus, as shown in Figure 2, the Pauli enhancement factor for TF(0.6)W is always positive, which could explain its relatively good performance. Nevertheless, the tendency of Fps,λ to become rather small and have high slope at high s may limit its performance in particular for semiconductors that are characterized by larger values of s. To overcome this problem, a new class of functionals (named Pauli−Gaussian, PGμ) can be constructed considering the positive-defined Pauli enhancement factor 2

Fsp(s) = e−μs ≥ 0

(7)

When μ ≈ 1.96λ − 5.33λ + 3.37, ≈ for 0 ≤ s ≤ 1. As expected, the PGμ functionals are significantly better than TFλW for semiconductors and have similar accuracy for simple metals, as shown in Figure 1. The best global performance is obtained approximately with μ = 1 (which defines the PG1 functional). However, this functional does not satisfy the second-order gradient expansion (GE2), which is instead an important exact property to be retained.53 On the other hand, the functional with μ = 40/27 ≈ 1.48 (named PGS, from Pauli−Gaussian second order) satisfies the GE2 constraint and is very good for semiconductors but quite bad for metals (see Figure 1). While we do not exclude that further optimizations of Fps (s) could lead to improved results, here we follow a different path, and we move to the Laplacian-meta-GGA level of theory, considering the PGSLβ (from Pauli−Gaussian second order and Laplacian) class of functionals, defined by 2

Fps,PGμ

2

Fps,λ

Figure 3. Linear response function of noninteracting HEG (π2χ(η)/ kF), computed from various KE functionals.

response function for different functionals. We see that TF(0.6)W and PG1 are good for η > 1, while PGS and VT84f are good for η < 1: thus, none of the gradientdependent functionals considered is able to reproduce the exact behavior for all η. On the other hand, a significant improvement is obtained by moving to the Laplacian level of theory: the PGSL0.25 functional describes accurately both the low- and the high-η regions. In more details, in Table 1, we report the accuracy of the various functionals for all of the considered properties, separately (results for all systems are reported in the Supporting Information). For simple metals, all of the semilocal functionals introduced in this Letter give quite accurate results, being comparable to the nonlocal KE functionals. On the other hand, semiconductors are more difficult systems, and all of the MAREs are larger ((but the errors for V0, which are comparable). Among the semilocal functionals considered, only PGSL0.25 gives always consistently accurate results. In particular, it performs better than 1/ 2(SM + HC) for both lattice constants and bulk moduli; for energies and densities (the hardest test), it is twice worse than 1/2(HC + SM) but still much more accurate than any other semilocal functionals. (A comparison of GaAs densities is reported in the Supporting Information.) Finally, in order to verify the broader applicability of the PGSL0.25 functional, we considered additional systems and properties. In Table 2, we report the MAREs for several silicon phases (sc, fcc, bcc, and cd, for cubic diamond) and vacancy formation energies for fcc Al, hcp Mg, and bcc Li.41 Again, for all properties, PGSL0.25 is the best semilocal functional, being always competitive with the nonlocal HC and SM functionals. In particular, PGSL0.25 performs well in the case of the vacancy formation energies of metals, which is a severe test for most KE functionals (here HC has a MARE of 88% and it also incorrectly predicts a negative Al-fcc vacancy formation energy; for all results, see the Supporting Information). In conclusion, we have shown that it is possible to achieve a realistic description of the KE of both metals and semiconductors at the Laplacian semilocal level of theory, without system-dependent parameters. This is an important result in view of future OF-DFT applications on large and complex

2

Fsp(s , q) = e−40/27s + F(q) ≈ e−40/27s + βq2 + ...

(8)

In eq 8, F(q) represents a generic function of the reduced Laplacian, which can be expanded in Taylor series because q is always small in solids. The linear term in q does not contribute to the energy nor to the potential;27 thus, the q2 term considered here is just the lowest-order correction. Note that the q2 term in eq 8 would cause divergence near the cusp of the electronic density (i.e., at the nuclei). However, this shortcoming is not present using pseudopotentials. In the tail of an exponentially decaying density (e.g., far away from an atom or a surface), q is diverging, but the full kinetic contribution τTFq2 is still exponentially decaying, being integrable.54,55 We also recall that the HEG fourth-order gradient expansion has been successfully applied to metallic clusters in the OF-DFT context.55 To fix the β coefficient in a nonempirical way, we could note that by setting β = 8/81 ≈ 0.1 in eq 8 the corresponding functional recovers the fourth-order linear response of the noninteracting HEG, i.e., the Lind4 functional; see ref 56. As shown in Figure 1, the Laplacian plays an important role. Varying β from 0 to 0.25, we observe a large improvement of simple metal properties (the RMARE decreases from 3 to 1.2) and correspondingly a very small change in the accuracy of semiconductors (RMARE increases from 0.9 to 1.2). The functional PGSL0.25 (β = 0.25) is competitive with HC and SM, being only 20% worse than their average. This is the second main result of this work: a very simple Pauli KE functional (a sum of an exponential and a Laplacian-dependent quadratic term) is almost as accurate as complicated and sophisticated nonlocal expressions. The good performance of the PGSL0.25 functional can be rationalized considering the exact density−density response function of the HEG system, which is 4387

DOI: 10.1021/acs.jpclett.8b01926 J. Phys. Chem. Lett. 2018, 9, 4385−4390

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The Journal of Physical Chemistry Letters

Table 1. MAREs (in %) for Equilibrium Volumes (V0), Bulk Moduli (B), Total Energies (E0), and Density Errors (D0) of the Simple Metals and Semiconductors Test Setsa simple metals TFW VT84f TF(0.6)W PG1 PGSL0.25 1/2(HC + SM) SM HC

semiconductors

V0

B

E0

D0

V0

B

E0

D0

4.8 *4.4 *2.6 *3.6 *4.4 4.3 3.4 5.0

20.3 15.8 12.1 *7.3 *7.4 7.8 *4.2 11.5

0.91 *0.14 *0.35 0.45 *0.35 0.34 0.21 0.47

4.4 3.9 3.1 3.7 2.8 1.7 *1.4 2.0

5.7 10.5 *2.9 *3.1 *2.5 4.5 7.6 *1.6

30.8 63.5 *9.2 *8.2 *5.7 21.1 37.7 *4.5

5.35 3.56 2.99 2.03 1.62 0.63 0.80 *0.46

23.4 22.1 16.1 14.7 13.3 7.9 *7.6 8.3

a

The best results from semilocal functionals are shown in bold, and a star indicates accuracy comparable to or better than 1/2(HC + SM).

energy−volume points, fitted using a birch-Murnaghan's equation of state61 expanded to 6th-order. The reference KS density was computed using the Abinit62 program. In this case, denser grids were used (10 grid-points per angstrom) .

Table 2. MAREs for Equilibrium Volumes (V0), Bulk Moduli (B), Total Energies (E0), and Density Errors (D) for Si Phases (sc, fcc, bcc, and cd)a TFW VT84F PG1 PGSL0.25 HC SM

V0

B

E0

D0

Evac

9.48 6.51 2.56 2.94 5.51 5.11

103.24 86.49 36.0 18.13 *10.41 19.76

2.28 0.94 0.69 0.53 0.28 *0.21

12.9 8.9 6.1 5.4 4.4 *2.6

133.8 74.5 52.2 30.5 88.5 *25.1

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b01926.

a

Full results for simple metals, semiconductors, Si phases, and vacancies and plot of the self-consistent electron densities of GaAs (PDF)

The last column reports MAREs for vacancy formation energies for fcc Al, hcp Mg, and bcc Li. The best results from semilocal functionals are shown in bold, and a star indicates the best functional.

■

systems (e.g., hybrid interfaces). Moreover, semilocal functionals of the type considered here can also easily be implemented in any real-space or plane-wave code. Finally, new developments can be considered starting from the present work. A first step concerns further optimization of the F(q) function in eq 8 and/or considering more complicated functional products of both s and q. To this end, systems with stronger inhomogeneity (i.e., larger q values) will be needed. A natural next step is thus extension of the proposed functional to interfaces or finite systems. Preliminary calculations carried out on molecular dimers have indeed shown that the PGSL0.25 functional can describe rather well the equilibrium bond length of dimers,34 with an accuracy similar to that of HC. Nevertheless, for further applications in this context, more careful treatment of exponentially decaying density regions (where both s and q diverge) must be considered. After this work was completed, we acknowledged development of the LKT GGA functional.58 The LKT functional form is numerically close to the PG0.75 one (for s < 1). An extensive assessment of the various KE functionals will be presented in a forthcoming article.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Lucian A. Constantin: 0000-0001-8923-3203 Eduardo Fabiano: 0000-0002-3990-669X Fabio Della Sala: 0000-0003-0940-8830 Notes

The authors declare no competing financial interest.

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COMPUTATIONAL DETAILS All calculations were performed using the PROFESS 3.0 code.59 For a better comparison with literature results, we chose the Perdew−Burke−Ernzerhof (PBE) XC functional38 for simple metals41 and the Perdew and Zunger XC LDA parametrization60 for semiconductors.36 We used bulk-derived local pseudopotentials (BLPSs), as in refs 36 and 41 and plane wave basis KE cutoffs of 1600 eV. Equilibrium volumes and bulk moduli were calculated by expanding and compressing the optimized lattice parameters by up to about 30% to obtain 20 4388

DOI: 10.1021/acs.jpclett.8b01926 J. Phys. Chem. Lett. 2018, 9, 4385−4390

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